The Briton started on pole and managed to come out on top of a sensational three-way battle with current title leader Kimi Antonelli and seven-time world champion Lewis Hamilton.
Norris crossed the line 11.54 seconds ahead of Antonelli in second place and claimed a memorable hat-trick for McLaren at the Dutch Grand Prix after wins for teammate Oscar Piastri last year and the Briton himself the year before.
"I made it tough for myself today, but a good race. Well fought. Probably one of my better wins. We had to fight for it well, but the pace was very strong," said Norris.
"I managed things incredibly well today. So over the moon, very, very happy and nice to win the final race here in Zandvoort," added the 26-year-old.
Mercedes driver George Russell completed the podium after holding off a determined challenge from Hamilton in the closing stages of the race.
A hefty downpour an hour before lights out meant drivers had to deal with a drying track and the conditions created a dramatic start to the race.
Coming into the last corner of the first lap, Verstappen lost control of his Red Bull, smashed into the wall and careered across the track, forcing several rivals to swerve out of the way.
To the relief of the tens of thousands of orange-clad Dutch fans, Verstappen immediately told his team he was unhurt and he walked free from the car.
When the race restarted after a delay of more than 30 minutes, Antonelli reacted faster off the grid than Norris, passing him in the first turn.
Antonelli was ahead until the 54th lap when Norris pulled off a spectacular overtake on the Italian.
Norris then cruised towards the chequered flag while behind him Hamilton engaged in a nail-biting scrap with George Russell, which the Mercedes man won.
The second Ferrari of Charles Leclerc came in fifth while the other McLaren driven by Oscar Piastri took sixth place.
Antonelli still enjoys a 59-point lead over rivals Hamilton and Russell, but he will start the next race in Monza with a 10-place grid penalty after Mercedes have been forced into an engine change.
"Just didn't have the feeling with the car that was sliding a lot overall," said Antonelli, who turns 20 next week.
It was the last edition of the Dutch Grand Prix, with promoters pulling the plug this year for financial reasons.
Drivers and fans alike regret the loss of the track, whose unusual layout with steep banked corners provides a unique challenge for the drivers.
Many drivers also hailed the raucous atmosphere at the event, with packed crowds -- most dressed from head to toe in Dutch orange -- ensuring a party spirit.
